Market Imperfections and Crowdfunding�

Anton Migloyand Victor Migloz

2017 This is a preliminary draft of an article published in SmallBusiness Economics Journal. The �nal version is available online

at: https://doi.org/10.1007/s11187-018-0037-1

Abstract

This paper o¤ers a model of crowdfunding that represents a growingarea of interest among practitioners and theorists. It is one of the �rstarticles analyzing the choice between the di¤erent types of crowdfunding(reward-based vs. equity-based) and the choice between crowdfundingand traditional �nancing. The model is based on standard market im-perfections such as asymmetric information and moral hazard as well ason some speci�c features of crowdfunding including the market feedbackregarding new projects. The model provides several implications, most ofwhich have not yet been tested. For example, we �nd that when asym-metric information is important, high-quality projects prefer reward-basedcrowdfunding. The choice of an all-or-nothing mechanism as opposed toa keep-it-all can serve as a signal of a �rm�s quality ("signalling by risk-bearing"). Crowdfunding is selected over a traditional bank loan if thedemand for the product is either very small or very large.

Keywords: crowdfunding, asymmetric information, moral hazard, equity-based crowdfunding, reward-based crowdfunding

JEL Codes: D82, G32, L11, L26, M13

1 Introduction

Crowdfunding is the practice of funding a start-up company or project by raisingfunds from a large number of people. It is usually performed on-line. The volumeof funds raised using crowdfunding has been quickly growing the last 5-7 years.In 2009 the volume of funds raised using crowdfunding was negligeably small.

�This article previously circulated as "Crowdfunding: Balancing Imperfect Informationand Moral Hazard Considerations". We are grateful to Peter Klein, the participants of RoyalEconomic Society annual conference and anonymous referees for the very helpful comments.Also, many thanks to Jason Pavunkovic, Kory Lippert, Alia Raza, Shane Smith, MichaelKidd, Jamie Grasman, Jonathon Dean, Melissa Toner, Erin Clark, and all the participants ofthe numerous discussions on crowdfunding organized by www.journalofcapitalstructure.comwebsite for their comments.

yBirmingham City University. anton.miglo@bcu.ac.uk.zUniversity of Toronto, v.miglo@mail.utoronto.ca

1

In 2014, crowdfunding platforms raised $16.2 billion, which is an increase fromthe $6.1 billion raised in 2013. Crowdfunding has more than doubled sincethen and raised $34.4 billion in 2015. Some analysts predict that crowdfundingmarket size will grow at an annual rate of 27.8% and will surpass venture capitalinvestments in the near future.1 Kickstarter, which is the leading crowdfundingplatform in the US, has raised over $2.4 billion in pledges from 10.9 millionbackers to fund almost 107,000 creative ideas.2

Crowdfunding research is quickly growing.3 As we are writing this article,the number of empirical papers signi�cantly exceeds the number of theoreticalpapers. Empirical papers on crowdfunding have found the following: crowd-funding relaxes geographic constraints on fundraising, which inhibit venturecapital and angel �nancing (Agrawal et al. (2010)); asymmetric informationand signalling seem to play a signi�cant role in crowdfunding (Ahlers, Cum-ming, Guenther, and Schweizer (2015), Hildebrand, Puri, and Rocholl (2014));success of a project and its delays are related to the volume of �nancing it re-ceives (Mollick (2014)); the timing of contributions usually follows a pattern(Kuppuswamy and Bayus (2015b)). Yet, the literature still lacks a full under-standing of how entrepreneurs choose between di¤erent types of crowdfundingand how they decide whether to use crowdfunding or other types of �nancing. Inthis paper we try to shed some light on these questions. We build a model thataddresses some of the aspects of crowdfunding mentioned above. In addition,our model incorporates other major features of crowdfunding. For example,in the case of crowdfunding the market provides intense feedback regarding a�rm�s projects and products. Unlike venture capital and bank �nancing, thereis no major investor with crowdfunding, who often maintain a certain degree ofmonitoring and control over the �rm�s activities. As a result, the entrepreneurmay be subject to a higher dregree of moral hazard (Agrawal, Catalini, andGoldfarb (2013), Moritz and Block (2014), Strausz (2017)). Finally, the ownersmay have better information about the quality of their products and their costs.We focus on the two leading types of crowdfunding: reward-based crowd-

funding (used by Kickstarter-the leading platform in the area) and equity-basedcrowdfunding. In the case of reward-based crowdfunding, investors count onsome extra-bene�ts from the company such as future product discounts. Underequity-based crowdfunding investors will receive shares of the company. Reward-based crowdfunding campaigns are commonly o¤ered in one of two models. The�Keep-It-All�(KIA) model involves the entrepreneurial �rm setting a fundrais-ing goal and keeping the entire amount raised, regardless of whether or not theymeet their goal, thereby allocating the risk to the crowd when an underfundedproject goes ahead. The �All-Or-Nothing�(AON) model involves the entrepre-

1See, for example, Salman (2016) or a global crowdfunding report onhttp://crowdfundbeat.com/2016/02/03/report-global-crowdfunding-market-2016-2020/

2Kickstarter website (June 1, 2016):https://www.kickstarter.com/help/stats?ref=about_subnav

3Moritz and Block (2014) and Kuppuswamy and Bayus (2015a) provide a review of theliterature in this �eld. For international aspects of crowdfunding see, for example, Gabison(2015), Miglo (2017), or Hat�eld (2017).

2

neurial �rm setting a fundraising goal and keeping nothing unless the goal isachieved, thereby shifting the risk to the entrepreneur. Kickstarter follows an�all or nothing�or threshold model, so funders�pledged money is only collectedif the goal is reached. While other crowdfunding e¤orts do not always followthis model, it is currently the dominant approach to crowdfunding, and parallelsthe way that other funding e¤orts for new ventures work. Our model is also re-�ective of the fact that crowdfunding is an area where production decisions and�nance are closely connected. The crowdfunding method choice directly andindirectly a¤ects the development of a project and its promotion, productionscale and price decisions.In our model each type of �nancing has its cost and bene�ts. Under reward-

based crowdfunding, it is harder to achieve the fundraising goal with largeprojects since the funders�potential bene�ts do not include the �rm�s long-termpro�ts unlike under equity-based crowdfunding. On the other hand, the en-trepreneur�s stake of equity is reduced under equity-based crowdfunding, whicha¤ects pricing and production decisions. In particular, we �nd that in this case,prices are higher and production quantities are lower than optimal since theentrepreneur receives less than 100% of the bene�ts from increasing productionwhile bears a non-shared extra-cost, therefore, the entrepreneur chooses a lowerlevel of production. We also �nd that high-quality projects are likely to chosereward-based crowdfunding as a signal of quality. Also, they are more likely tobe funded through the AON scheme. Low-quality or high-risk projects are lesslikely to mimick high-quality �rms and chose AON, which implies more fundras-ing responsability and risks, and prefer KIA instead. Traditional bank �nancingmay lead to bankruptcy if the �rm is unsuccessful. So the magnitude of thebankruptcy cost plays a role in the �nancing method choice. If these costs arehigh enough, the entrepreneur may prefer crowdfunding since, formally, crowd-funding does not neccessarily lead to bankruptcy if the crowdfunding campaignor production fails. However, under reward-based crowdfunding, indirect costsof distress may arise related to consumer protection law in case products arenot delivered to customers. We �nd that a separating equilibrium where high-quality �rms select reward-based crowdfunging can only exist if these costs arerelatively high. Finally, unlike traditional �nancing, crowdfun�ng provides mar-ket feedback. When this feature of crowdfunding is introduced into the basicmodel, we �nd that crowdfunding is selected over a traditional bank loan if thedemand for the product is either very small or very large.As was mentioned previously, the number of theoretical papers on crowd-

funding is relatively small. Note the following. Belle�amme, Lambert, andSchwienbacher (2010) identify a number of issues related to crowdfunding froman industrial organization perspective. In their model, they analyze reward-based crowdfunding with pre-ordering and price discrimination, and study theconditions under which crowdfunding is preferred to traditional forms of exter-nal funding. In the second model, crowdfunding is a way to make a productbetter known to consumers. The authors argue that non-pro�t organizationstend to be more successful in using crowdfunding.Belleammey, Lambertz and Schwienbacher (2014) compare reward-based

3

and equity-based crowdfunding. In either case, the funders enjoy communitybene�ts that increase their utility. It is shown that the entrepreneur preferspre-ordering if the initial capital requirement is relatively small compared tothe market size and prefers pro�t sharing otherwise. Belleammey et al (2014)also o¤er some extensions on the impact of quality uncertainty and informationasymmetry but in these extensions the choice between di¤erent forms of crowd-funding and other forms of �nancing is not modelled. As the authors mentioned,further research is required.Strausz (2017) studies entrepreneurs�interactions with customers before in-

vestment using the mechanism design approach. Under aggregate demand un-certainty, crowdfunding improves the screening of potential customers. Entre-preneurial moral hazard threatens this bene�t. Studying the subsequent trade-o¤ between screening and moral hazard, the paper characterizes optimal mech-anisms. E¢ ciency is sustainable only if returns exceed investment costs by amargin re�ecting the degree of moral hazard. Constrained e¢ cient mechanismsexhibit underinvestment.Hu, Li, and Shi (2014) study the optimal product and pricing decisions in a

crowdfunding all-or-nothing mechanism. When the buyers are su¢ ciently het-erogenous in their product valuations, the creator should o¤er a line of productswith di¤erent levels of product quality. Compared to the traditional situationwhere orders are placed and ful�lled individually, with the crowdfunding mech-anism, a product line is more likely to be optimal than a single product and thequality gap between products is smaller. The paper also shows the e¤ect of thecrowdfunding mechanism on pricing dynamics over time. Together, these resultsunderscore the substantial in�uence of the emerging crowdfunding mechanismson common marketing decisions.The rest of the paper is organized as follows. Section 2 describes the basic

model and some preliminary results. Section 3 through 6 discuss the conse-quences of introducing di¤erent kinds of market imperfections into the basicmodel and their implications for crowdfunding decisions. Section 7 analyzescases that involve several market imperfections simultaneously. Section 8 dis-cusses the consistency of the model�s predictions with observed empirical evi-dence. Section 9 discusses the model�s robustness and its potential extensionsand Section 10 is a conclusion to the study.

2 Basic Model

An entrepreneurial �rm has monopoly power over its innovative product or ser-vice. The �xed costs of launching the production equal I. The �rm intendsto sell its product in two consecutive periods. In period t = 1; 2, if the �rmproduces qt units, it costs cqt in total. The demand for the good in each periodis given by the inverse demand function pt = a � qt.4 The �rm needs funds to

4Some papers use the approach where there are individual customers with di¤erent demandfunctions (see, for example, Belleammey et al (2014) and Hu, Li and Shi (2014)) or wherethere is a possibility of product substitution between periods. Section 9 discusses the model�s

4

cover its start-up costs and is considering crowdfunding. Under reward-basedcrowdfunding the �rm collects pre-orders for period 1.5 Under equity-basedcrowdfunding, the �rm sells a fraction � of the �rm. Funders and entrepreneursare assumed to be risk-neutral and the risk-free interest rate is 0. A two-periodmodel will help us understand the di¤erence between basic features of di¤er-ent types of crowdfunding (like long-term character of earnings in the case ofequity-based crowdfunding vs. short-term rewards in the case of reward-basedcrowdfunding) as well as capture some other important features of crowdfund-ing such as incorporating market feedback during period 1 into the productquality. Since crowdfunding is usually used to cover the start-up costs, period2 �nancing is not explicitly modelled. The capital structure and the ownershipstructure will remain the same in period 2 as they are at the end of period 1.Earnings will be distributed accordingly.

2.1 Reward-Based Crowdfunding: Pre-orders

The timing of events is as follows:

1. Firm selects p1 (pre-order price). The demand for the product is de-termined. If p1q1 < I + cq1, the �rm is liquidated.6 Otherwise, theentrepreneur collects pro�t (p1 � c)(a� p1)� I.

2. Firm selects p2. The entrepreneur collects pro�t (p2 � c)(a� p2).

In this setting, the �rm selects a pre-order price in order to maximize itspro�ts. The constraint, however, comes from the necessity to collect the amountof money required to launch production.In period 2, the �rm chooses p2 to maximize (p2 � c)(a � p2), which gives

p2 =a+c2 .

In period 1, the �rm maximizes (p1 � c)(a � p1) � I subject to: p1q1 =p1(a� p1) � I + cq1 = I + c(a� p1). This condition means that the amount ofpre-orders should cover the start-up cost (�xed costs and the period 1�s variablecosts).Two cases are possible. If

(a� c)24

� I (1)

robustness with regard to changes in the demand functions and other features of the model.5Existing studies consider consumer �nancing through pre-ordering, bootstrap �nancing

(see, e.g., Winborg and Landstrom, 2001; and Ebben and Johnson, 2006) or working capitalloans. However, they do not usually distinguish between advance payments made at the verybeginning of an entrepreneurial initiative and those made during the course of further devel-opments. Crowdfunding pertains speci�cally to the �nancing of innovative entrepreneurialprojects. There are many features of this type of �nancing such as market feedback froma large number of funders, which is typically not included in existing studies on consumer�nancing or bootstrap �nancing etc.

6The presence of thresholds for the minimum required amount of funds (the poject fails ifthe thresholds are not met) is typical in crowdfunding. We begin our analysis with a naturalassumption as to why this occurs: a �rm�s inability to cover the start-up costs. In Section 5we will discuss other reasons for possible thresholds.

5

then p1 = a+c2 .

The �rm�s pro�t over the two periods equals

� =(a� c)24

� I + (a� c)2

4=(a� c)22

� I (2)

If (1) fails, the �rm will not be able to raise the funds needed to launchthe production. When the required amount of initial investment is quite large,reward-based crowdfunding may not be an option.

2.2 Equity-Crowdfunding: Pro�t-Sharing

The timing of events is as follows:

1. Firm selects � (the fraction of the �rm for sale) and p1 and sells � forprice M . If M < I + cq1, the �rm is liquidated.

2. Firm selects p2.

In this setting, the �rm has more �exibility in rasing the initial amount ofinvestments, since the funders can also count on the second period�s (future)pro�t.In period 2, the �rm chooses p2 to maximize the entrepreneur�s pro�t (1 �

�)(p2�c)(a�p2), which makes p2 = a+c2 . The �rm�s pro�t in period 2 is

(a�c)24 .

In period 1, the �rm chooses � and p1 to maximize the entreprneur�s ex-pected pro�t over the two periods:

(1� �)(p1(a� p1) +M � I � cq1 +(a� c)24

) (3)

subject toM � I + cq1 (4)

The funders�expected earnings should cover their investment cost or:

�(p1(a� p1) +(a� c)24

) �M (5)

For the optimal solution the conditions (4) and (5) will be binded becausethe �rm can always make � as small as necessary to satisfy them. Then wehave:

� =I + cq1

p1(a� p1) + (a�c)24

(6)

Substituting this into (3) makes the entrepreneur�s expected pro�t over the twoperiods equal to:

(p1 � c)(a� p1)� I +(a� c)24

This implies p1 = a+c2 .

6

The entrepreneur�s expected pro�t then equals

(a� c)24

� I + (a� c)2

4=(a� c)22

� I (7)

As we can see, it is the same amount as in (2). This is not surprising giventhat in the absence of any �nancial market imperfections every type of �nancingshould have the same result (similar to Modigliani-Miller proposition (1958)) aslong as they �t into the budget constraints.Lemma 1. If I is su¢ ciently small ( (a�c)

2

4 � I), the �rm is indi¤erentbetween reward-based and equity-based crowdfunding. If I is large, equity-basedcrowdfunding is preferred.The proof of this lemma follows from the above analysis. If I is small, the

�rm�s pro�t is the same under the two types of crowdfunding ((2) and (7)). If Iis large, it follows from the previous subsection that the �rm is not able to raiseenough funds to cover its start-up costs using a reward-based crowdfunding.Lemma 1 shows that equity-based crowdfunding has a "technical" advantage

for large projects (high �xed costs I and high variable costs c). Since our focusis on the role of market imperfections, we will usually assume that condition (1)holds in the further analysis, i.e. both types of crowdfunding are feasible.

3 Moral hazard: costly entrepreneurial e¤ort

So far we assumed that the decisions about � and p1 are made simultaneously.We know, however, that under equity-based crowdfunding, the entrepreneur�sshare of the company is less than 100% after funds are raised and therefore theentrepreneur�s incentive may be di¤erent than it would be under reward-basedcrowdfunding.7 Hence, we consider a situation where the cost of productionalso includes the entrepreneur�s own e¤ort. We assume that this e¤ort costs eq.Following similar calculations to those in the previous subsection, one can seethat under reward-based crowdfunding p1 = p2 = a+c+e

2 and the entrepreneur�spro�t equals

(a� c� e)22

� I (8)

Under equity-based crowdfunding the results may be di¤erent because of theentrepreneurial moral hazard resulting from the reduced equity stake.The timing of events is as follows:

1. Firm selects � and sells it for price M .

2. Firm selects p1.

3. Firm selects p2.

7This is a classical moral hazard idea (Jensen and Meckling, 1976).

7

Proposition 1. 1) If (a�c�e)22 � I, the �rm prefers reward-based crowd-

funding; 2) Prices are higher and the quantity produced is lower under equity-based crowdfunding than under reward-based crowdfunding.Proof. See Appendix.As shown in the Appendix, p1 = p2 =

a+c+e�(1��)2 . Under equity-based

crowdfunding, the price is higher than it is under reward-based crowdfunding.This is intuitive because the entrepreneur reaps less than 100% of the bene�tsfrom increasing production while bears a non-shared extra-cost, therefore, theentrepreneur chooses a lower level of production.It is also shown that the entrepreenur�s pro�t over the two periods equals

(a� c)22

� I � e2

2(1� �)2 +e2

1� � � ea+ ec (9)

If � = 0, (9) will be equal to (a�c�e)22 � I. It was mentioned above that it

would be the same value as it would be in the case of reward-based crowdfunding.When � is positive, the entrepreneur�s pro�t under equity crowdfunding will besmaller since the derivative of (9) in � is negative. It is consistent with the ideaof agency cost.

4 Asymmetric information about cost

So far we assumed that investors have the same information as entrepreneurs.Now suppose that the �rm can be either a low-cost (high-e¢ ciency) producer(denoted l) or a high cost (low-e¢ ciency) producer (denoted h). More speci�-cally, suppose that c is either equal to cl or ch and cl < ch. Initially the �rm�stype (the value of c) is determined and becomes known to the entrepreneur.The timing of events is as follows:

1. The �rm�s type is revealed to the entrepreneur.

2. Firm selects �nancing strategy: reward-based crowdfunding or equity-based crowdfunding.

3. If equity-based crowdfunding is selected, � is determined and the �rm sellsit for price M . If M < I + cq1, the �rm is liquidated.

4. Firm selects p1.

5. Firm selects p2.

An equilibrium is de�ned as a situation where no �rm type has an incentive todeviate. Since private information only concerns the production cost and not thedemand side, the informational game will only a¤ect the equity-crowdfundingscenario.8 The price that potential investors will be paying for a fraction of

8We mostly focus on separating equilibria since it generates meaningful empirical impli-cations. Further reserach is required regarding the exisitence and implications of poolingequilibria.

8

a �rm�s shares depends on their beliefs about the �rm�s production cost. Theinformation game does not a¤ect the outcome of reward-based crowdfunding.Firms will select their prices as in the case with perfect information and demandwill be determined by the demand functions that are publicly known in thisscenario.9 This leads to the point that if a separating equilibrium exists, it willnot be one where the high-e¢ ciency type chooses equity-based crowdfundingsince it will always be mimicked by the low-e¢ ciency type. This result is typicalfor basic models with asymmetric information beginning with Akerlo¤ (1970).

Proposition 2. If I is su¢ ciently small ( (a�cl)2

4 � I), an e¢ cient sep-arating equilibrium exists, where type l selects reward-based and type h selectspro�t-sharing. An e¢ cient separating equilibrium where type h selects reward-based and type l selects pro�t-sharing does not exist.Proof. See Appendix

5 Asymmetric information about demand

In this section, asymmetric information concerns the quality of a �rm�s productsand services. In particular, we assume that, unlike outside investors, �rm ownersknow the value of parameter a in the demand function. In the setup discussedin the previous section, a low-quality �rm will always have an incentive tomimick a high-quality �rm when the latter uses equity-based crowdfunding.Intutively, a similar engine should drive the results if the asymmetric informationregards the product�s quality rather than its cost. In order to obtain new results,we introduce new strategies. In particular, if the �rm selects reward-basedcrowdfunding, it has two options: KIA (keep-it-all) or AON (all-or-nothing). IfAON is selected, a threshold T is set, T > 0. If the amount of funds raised inperiod 1 is less than T , the �rm is liquidated. We also assume that the demandis as follows: qt = �t(a � pt), where �t = 1 with probability � and 0 withprobability 1��. Making the demand function stochastic or risky will allow usto see the role of AON method of crowdfunding ("signalling by risk-bearing").10

Note that some empirical research suggests that many crowdfunding projectsattract very low or negligeably small amounts of funds (see, for example, Mollick(2014), Cordova, Dolci and Gianfrate (2015) and Desjardins (2016)). �1 becomesknown after the project is created and the crowdfunding method is selected. �2becomes known in the beginning of period 2. Also, we assume that there aretwo types of �rms: a = ah for type h and a = al for type l, where ah > al. Tofocus on the e¤ect of asymmetric information, we assume, I = 0 (none of theresults change qualitatively if I > 0). In particular it implies that condition (1)holds for both types of �rms meaning they can use reward-based crowdfunding.Also, it means that a �rm should follow the rule p > c in order to accumulate

9 In the next section we will consider a situation with asymmetric information about de-mand.10For simplicity, previous sections did not di¤erentiate between the di¤erent types of reward-

based crowdfunding. One can easily check that it would not a¤ect the results. The same holdswith the assumption about the stochastic demand function.

9

su¢ cient funds to launch their product and avoid liquidation.The timing of events is as follows:

1. The �rm�s type is revealed to the entrepreneur.

2. Firm selects �nancing strategy: KIA, AON or equity-based crowdfunding.If AON is selected, the �rm selects T .

3. �1 becomes known.

4. If equity-based crowdfunding is selected, the �rm selects � (fraction ofshares) and sells it for an amount M .

5. Firm selects p1.

6. If AON is selected and p1q1 < T , the �rm is liquidated.

7. If KIA or AON and p1q1 < cq1, the �rm is liquidated. If equity-basedcrowdfunding is selected and M < cq1, the �rm is liquidated.

8. Firm�s type (product�s quality) becomes publicly known.

9. �2 becomes known.11

10. Firm selects p2.

First consider the symmetric information case for KIA.In period 2, if �2 = 1 and q = a � p2, the �rm chooses p2 to maximize

(p2 � c)(a � p2), which makes p2 = a+c2 . If �2 = 0 and q = 0, the �rm�s pro�t

is zero.In period 1, if �1 = 1 and q = a � p1, the �rm maximizes (p1 � c)(a � p1).

We have p1 = a+c2 . If �1 = 0 and q = 0, the �rm�s pro�t is zero.

The �rm�s expected pro�t equals

� = �(a� c)24

+ �(a� c)24

=�(a� c)2

2(10)

Now consider AON. In this setting, the �rm selects the pre-order price in orderto maximize its sales. At the same time, it needs to reach the establishedthreshold amount of pre-orders. In some cases it will force the �rm to selecta suboptimal pricing policy and in some cases (when the initial investment issu¢ ciently large), the project will not be successful. Also, bankruptcy is unavoidable under AON, if the demand is zero.In period 2, if �2 = 1 and q = a � p2, the �rm chooses p2 to maximize

(p2 � c)(a � p2), which makes p2 = a+c2 . If �2 = 0 and q = 0, the �rm�s pro�t

is zero.11 In this section, the timing of information revelation about a product�s quality and demand

is the same for any type of crowdfunding. In Section 7, we analyze the di¤erences betweendi¤erent types of crowdfunding in terms of their ability to a¤ect product�s quality.

10

In period 1, the �rm chooses T and p1 to maximize � where � = �((p1 �c)(a� p1) + � (a�c)

2

4 ) if p1q1 = p1(a� p1) � T .� = 0 if p1q1 = p1(a� p1) < T .The solution is any T such as T � p1(a � p1) where p1 = a+c

2 . It does notavoid liquidation if demand is zero in period 1 but it optimizes the price policyif demand is positive.The �rm�s expected pro�t equals

� = �((a� c)24

+ �(a� c)24

) =�(1 + �)(a� c)2

4(11)

This is smaller than (10) because under AON, bankruptcy will occur inperiod 1 if the amount of raised funds is smaller than T .Finally, consider equity-based crowdfunding. In period 2, if �2 = 1 and

q = a � p2, the �rm chooses p2 to maximize the entrepreneur�s pro�t (1 ��)(p2 � c)(a � p2), which makes p2 = a+c

2 . If If �2 = 0 and q = 0, the �rm�s

pro�t is zero. The �rm�s expected pro�t in period 2 is �(a�c)24 .

In period 1, if �1 = 1 and q = a� p2, the �rm chooses � and p1 to maximizethe entrepreneur�s pro�t:

(1� �)(p1(a� p1) +M � cq1) (12)

subject toM � cq1 (13)

The funders�expected earnings over the two periods should cover their in-vestment cost or:

�(p1(a� p1) +�(a� c)2

4) �M (14)

Under the optimal solution the conditions (13) and (14) will be binded be-cause the �rm can always make � as small as necessary to satisfy them. Thenwe have:

� =cq1

p1(a� p1) + �(a�c)24

Substituting this into (12) and using the fact that if �1 = 0 and q = 0, shares arenot sold and the �rm�s pro�t in period 1 equals 0, we �nd that the entrepreneur�sexpected pro�t over the two periods equals:

�(p1 � c)(a� p1) +�(a� c)2

4

This implies that p1 = a+c2 .

The entrepreneur�s expected pro�t then equals

�(a� c)24

+�(a� c)2

4=�(a� c)2

2(15)

As we can see, this is the same amount as in (10).

11

Now consider asymmetric information.Proposition 3. If ( al�cah�c )

2 < � < 2 � (ah�cal�c )2, a separating equilibrium

exists, where type l selects keep-it-all and type h selects all-or-nothing or equity-based crowdfunding. An equilibrium where type h selects keep-it-all or equity-based crowdfunding does not exist.Proof. See Appendix.The right side of the inequality in Proposition 3 puts an upper bound on

the probability of bankruptcy. The intuition behind this result is as follows.AON is very costly if the probability that the demand is absent is relativelyhigh. In this case the low-quality �rm will not mimick the high-quality �rm. If,on the contrary, � is very large, the values of (10), (11) and (15) do not di¤ersigni�cantly for the low-quality �rm (they are equal in the extreme case when� = 1) which means that the low-quality �rm would mimick the high-quality�rm and bene�t from the market�s optimistic belief about the quality of �rmsthat use AON. The left side of the inequality in Proposition 3 places a lowerbound on the probability of bankruptcy. If, on the contrary, the probabilitythat demand is absent is very high, it would be bene�cial for the high-quality�rm to not use AON and deviate to KIA or equity-based crowdfunding.

6 Bankruptcy costs and bank monitoring

In this section we compare crowdfunding with bank �nancing. If the �rm takesa bank loan and it is not able to pay back its debt then the �rm is bankrupt andcan be liquidated. On the other hand, banks have a better ability to monitor andcontrol entrepreneurs.12 So we assume that the manager (managerial team) hassome private bene�ts b from each unit produced at the expense of the �rm whenthe �rm uses crowdufunding. To simplify the calculations related to bankruptcywe assume that the production output is stochastic in period 1 and depends onparameter Q (similar to stochastic demand in Section 5): Q = 1 with probability and 0 with probability 1� . This implies that bankruptcy will only occur ifthe �rm takes a bank loan and Q = 0. In contrast to �rm liquidation cases whenthe required �nancing is not raised, bankruptcy does not occur as a result offailed production if the �rm uses crowdfunding.13 For simplicity assume I = 0.This implies that the condition (a�c�b)2

4 � I holds for both �rm types, whichimplies that crowdfunding is feasible for each type (similar to Section 3, formula(8)).The timing of events is as follows:

12See, for example, Diamond (1984).13 In most countries there is no formal regulation that can be used to force a company into

bankruptcy in the case of crowdfunding (see, for example, Gabison (2015) or Moores (2015)).There is di¤erence, however, between equity-based and reward-based crowdfunding. If the�rm uses reward-based crowdfunding then the consumers are under consumer protection lawetc. (Gabison (2015)). We consider this aspect in Section 7. Here we assume that in contrastto traditional bank �nancing there is no bankruptcy in the case of crowdfunding. For simlicityit is assumed that the �rm uses equity-based crowdfunding.

12

1. Firm selects �nancing strategy: bank loan or crowdfunding.

2. Firm selects p1.

3. Q becomes known. If Q = 0 and bank loan was selected, the �rm isbankrupt.

4. Firm selects p2.

Proposition 4. 1) Prices are higher and quantity produced is lower undercrowdfunding; 2) For given values of a and , there exists a b� such that the �rmchooses to take a bank loan if b � b� and chooses crowdfunding if b < b�. Forgiven values of a and b, the �rm chooses to take a bank loan if is su¢ cientlylarge.Proof. See Appendix.We �nd that the product price under bank �nancing is p = a+c=

2 and

p = a+(b+c)= 2 under crowdfunding. Prices are higher and quantity produced

is lower under crowdfunding because of the extra-cost related to moral hazardissues. The second part of the proposition states that crowdfunding will bepreferred if the cost related to the absence of monitoring is relatively small.Otherwise, a bank loan will be preferred. Interestingly, the e¤ect of a changein the probability of bankruptcy is not as straightforward as the e¤ect of b. Ifthe probability of bankruptcy is close to zero then a bank loan will de�nitely bepreferred because of the monitoring advantage. However, in the middle range ofthe values for , one may �nd that an increase in bene�ts crowdfunding morethan a bank loan. The reason for this follows from the price formulas above: asmall ampli�es the �rm�s moral hazard issues making the price further fromoptimal.

7 Hybrid cases

Ideally, the next step would be to analyze optimal �nancing policy when manyfactors such asymmetric information, moral hazard, market feedback etc. arepresent in the model simultaneusly. This is an intriguing challenge for futureresearch. One should say that the creation of such a universal global modelis technically di¢ cult and in many cases may not bring many analytical andintuitively sound results.14 This section provides an example of such an analysis.

14A good example is capital structure theory. Most intuitions published in textbooks forthe last 50 years are based on models that consider each factor separately (pecking ordertheory for asymmetric inofrmation, trade-o¤ thoery for taxes and bankruptcy costs etc.). Foran example of capital structure theory review and the role of market imperfections see Harrisand Raviv (1991), Miglo (2011) and Miglo (2016). Models combining several factors are muchless popular and much more technically complicated though some researchers suggest thatthese types of models are a prominnet direction for future research. Also note that basedon managers� surveys, managers only support around 50% (see, for example, Graham andHarvey (2001)) of basic theories, which means that the precentage of managers that use evenmore complicated ideas is even smaller. Crowdfunding theory is a much younger theory than

13

Case 1. Consider the situation where �rms have private information aboutproduction costs (Section 4). In this situation reward-based crowdfunding canbe used as a signal of a �rm�s quality. Now suppose that a �rm is terminated(bankrputcy occurs) in period 1, if the �rm is not able to deliver its productto customers and the �rm uses reward-based crowdfunding (similar to the ideasfrom sections 5 and 6). Gabison (2015) noted15 that eventhough there is noformal regulation of reward-based crowdfunding in most countries, in most casesconsumers (funders) are under consumer protection law (which exists in mostdeveloped countries) and therefore a violation of this law can be costly for the�rm. As in Section 4, c is either equal to cl or ch and cl < ch. Like in Section6, the production output is stochastic in period 1 and depends on parameterQ (similar to stochastic demand in Section 6): Q = 1 with probability or 0with probability 1� . Bankruptcy only occurs when Q = 0 and the �rm usesreward-based crowdfunding. Bankruptcy does not occur as a result of failedproduction in period 1 under equity-based crowdfunding since by its nature nopromises are made to funders/investors and dividends are not guaranteed. Likein Section 5, for simplicity we assume I = 0.The timing of events is as follows:

1. The �rm�s type is revealed to the entrepreneur.

2. Firm selects �nancing strategy: reward-based crowdfunding or equity-based crowdfunding.

3. If equity-based crowdfunding is selected, � is determined and the �rm sellsit for price M .

4. Firm selects p1.

5. If equity-based crowdfunding is selected and M < cq1, the �rm is liqui-dated. If reward-based crowdfunding is selected and p1q1 < cq1, the �rmis liquidated.

6. Q becomes known. If Q = 0 and reward-based crowdfunding was selected,the �rm goes bankrupt.

7. Firm selects p2.

An equilibrium is de�ned as a situation where no �rm type has the incentiveto deviate. Like in Section 4, since information only concerns the productioncost and not the demand side, the informational game will only a¤ect the equity-crowdfunding scenario.

capital structure theory so it is in the stage of its development where the quality and relativesimplicity of its basic ideas are probably the most important objectives of its research alongwith managerial education on these ideas (see, for example, Loane, Ramsey and Ibbotson(2016)).15See also Ibrahim (2016) and Moores (2015) for a legal environment analysis regarding

reward-based crowdfunding. Mollick (2015) empirically analyzes the percentage of failed �rmsthat used reward-based crowdfunding.

14

Proposition 5. If a�cla�ch < 2, a separating equilibrium does not exist. Oth-

erwise, if is su¢ ciently large, the only e¢ cient separating equilibrium thatexists is one where type l selects reward-based crowdfunding and type h selectspro�t-sharing.Proof. See Appendix.To explain the results of this proposition, note that Section 4 found that

high-quality �rms can use reward-based crowdfunding to signal their quality.That section did not consider a potential cost of reward-based crowdfunding re-lated to bankruptcy in the case when the �rm is not able to deliver their productin period 1. This case asks if the result stands if such a cost is taken into con-sideration. What we found is that the result stands but there are cases when aseparating equilibrium where a high-quality �rm uses reward-based crowdfuningdoes not exist. The meaning of the condition stated in the proposition is that ifthe di¤erence between the �rm types is su¢ ciently small, such an equilibriummay not exist. Secondly and more interestingly is that if the probability ofbankruptcy is su¢ ciently small, an equilibrium may not exist. In this case, alow-quality �rm may still be interested in mimicking a high-quality type whenthe latter choses reward-based crowdfunding.Case 2. Similar to some previous sections, this case considers a model with

imperfect information. However, here we assume that crowdfunding helps the�rm obtain information about demand. Suppose that if the �rm uses crowd-funding, it can improve the product�s quality after obtaining useful informationabout demand in period 1: more speci�cally, in period 2 the demand becomesq = sa � p; s � 1.16 We assume that s has di¤erent values for di¤erent typesof crowdfunding: s 2 fsr; seg ; sr > se where sr is the product improvement ifreward-based crowdfunding is used. sr > se because under reward-based crowd-funding, the funders know that the �rm�s launch of production and, respectively,its survival depend on their pre-orders and the �rm�s response to this feedbackis expected to be very e¢ cient since the �rm�s survival depends on it.17 Also,under reward-based crowdfunding, the funders have a short-term interactionwith the �rm whereas under equity-based crowdfunding, these interactions arelong-term. So the former incentivizes the funders to provide a more intensefeedback. If the �rm uses traditional �nancing like a bank loan, for example, itdoes not get the same feedback as it would with crowdfunding and the demanddoes not change in period 2. On the other hand, as in Section 6, banks have a

16Xu, Yang, Rao, Fu, Huang, and Bailey (2014), Block, Hornuf and Moritz (2016) and daCruz (2016) empiricaly analyze di¤erent aspects of the informational value of crowdfundingfor entrepreneurs.17Note that the market feedback represents probably the most important community bene�t

of crowdfunding for the �rm (because it may increase its product quality and repsectively theirfuture pro�ts) as well as for funders and customers who can enjoy higher quality products asa result of market feedback. Note also that we explicity model this mechanism in our modelthrough providing better information to the �rm in period 1, which allows them to improvetheir product�s quality in period 2 etc. Belleammey et al (2014) assume that there are someexogenously given community bene�ts in period 1 as a result of crowdfunding. Note thatCholakova and Clarysse (2015) �nd that non-monetary bene�ts do not play a singi�cant rolefor funders.

15

better ability to monitor and control the entrepreneurs.18 We assume that themanager has some private bene�ts b when using crowdfunding.The timing of events is as follows:

1. Firm selects a �nancing strategy: bank loan, reward-based crowdfundingor equity-based crowdfunding.

2. If equity-based crowdfunding is selected, the �rm chooses � (the fractionof the �rm for sale) and sells it for price M . If M < I + cq1, the �rm isliquidated.

3. Firm selects p1. The demand for the product is determined.

4. Firm selects p2. The demand for the product is determined.

Proposition 6. For a given value of a, if I is su¢ ciently small, the �rmtakes a bank loan if sr is su¢ ciently small or b is su¢ ciently large. Other-wise, the �rm selects reward-based crowdfunding. If I is su¢ ciently large,the �rm takes a bank loan if se is su¢ ciently small or b is su¢ cientlylarge. Otherwise, the �rm selects equity-based crowdfunding. Prices arehigher and quantity produced is lower under crowdfunding. For a givenvalue of I, crowdfunding is selected over a traditional bank loan if a iseither very small or very large. For medium levels of a, a bank loan ispreferred.

Proof. See Appendix.

It shown in the Appendix that the entrepreneur�s pro�ts under the di¤erentstrategies are equal to the following.

�r =(a� b� c)2

4+(sra� b� c)2

4� I

�e =(a� b� c)2

4+(sea� b� c)2

4� I

�b =(a� c)22

� I

where subscript r stands for reward-based crowdfunding, e means equity-based crowdfunding and b means bank loan.The �rm is indi¤erent between reward-based crowdfunding and a bank loan

if(a� b� c)2

2+(sra� b� c)2

2= (a� c)2 (16)

The �rm is indi¤erent between equity-based crowdfunding and a bank loanif

(a� b� c)22

+(sea� b� c)2

2= (a� c)2 (17)

18Other traditional forms of entrepreneurial �nancing such as venture capital �nancing alsohave a high degree of monitoring so the model can be applied to those cases as well.

16

Also if(a� c� b)2 < 4I (18)

the �rm will not be able to use reward-based crowdfunding. And if (a�c�b)2 �4I, the �rm prefers reward-based crowdfunding over equity-based crowdfunding.Figure 1 illustrates the equilibrium decision-making for the entrepreneurs.

The lines represent equations (16), (17) and (18). Letters RC, EC and Bdenote the areas where the entrepreneurs choose reward-based crowdfunding,equity-based crowdfunding, and a bank loan respectively.

-

6

I

a

B

EC

EC

(18)

(16)(17)

B

RC

RC

Figure 1. The choice of �nancing.

As follows from Figure 1, �rms that use crowdfunding are either projectswith very small demand or very high demand. Also, entrepreneurs with EChave higher I for any value of a compared to entrepreneurs with RC. Overallwe can see that �rms with a medium level of demand prefer B, �rms withstronger demand prefer crowdfunding, �rms with a large amount of investmentsand strong demand or very weak demand prefer EC and �rms with smallerinvestments and strong demand or very weak demand prefer RC.

8 Implications

Our paper has several implications for an entrepreneurial �rm�s choice of �nanc-ing. The summary of the results is presented in Table 1.

17

Market imperfection(s) ResultsAsymmetric information about produc-tion cost

Good quality projects prefer reward-based crowdfunding

Entrepreneur�s moral hazard due to hisreduced equity stake

Firms prefer reward-based crowdfunding.Prices are higher and quatity produced islower under equity-based crowdfunding

Asymmetric information about demand "Signalling by risk-bearing". Low-quality�rm selects KIA or equity-based crowd-funding and high-quality �rm selectsAON

Bankruptcy costs vs. bank monitoring Prices are higher and quantity producedis lower under crowdfunding than under abank loan

Hybrid case 1 (asymmetric informationabout demand and bankruptcy costs)

If bankrupcy costs are high, �rms usereward-based crowdfunding to signal theirquality

Hybrid case 2 (market feedback, bankmonitoring)

Prices are higher and quantity produced islower under crowdfunding. Crowdfundingis selected over a traditional bank loan ifdemand is either very small or very large.

Table 1. Market imperfections and the model�s results.Proposition 2, 3 and 5 imply that when asymmetric information is important,

high-quality projects prefer reward-based crowdfunding. This is contradictory,to some extent, to the spirit of the results in Belleammey et al (2014), which�nds that asymmetric information favors equity-based crowdfunding.19 Note,however, that the objective of their analysis is di¤erent from ours. For example,they do not analyze the case when the decision about the choice of crowdfund-ing mode is part of the model (this is obviously a crucial part of our model;consequently they automatically do not consider the possibility that �rms cansignal their quality with their choice of crowdfunding) so they only comparethe symmetric and asymmetric information cases within each type of crowd-funding. Also, it is mentioned in Belleammey et al (2014) that their analysis ofthe asymmetric information case is not complete.20 In our model, equity-basedcrowdfunding su¤ers more from asymmetric information, which is consistentwith the spirit of the majority of �nance literature where equity-�nancing is

19For example, it is well-known in capital structure theory that asymmetric informationdamages equity �nancing more than debt �nancing and that equity �nancing can not be usedby a high-quality type as a signal of quality whereas in some cases debt �nancing can be used(Leland and Pyle (1977)). So applying this example to Belleamey et al (2014), who claimthat asymmetric information is more damaging for reward-based crowdfunding, it would beno surprise to �nd that a separating equilibrium where a high-quality �rm uses reward-basedcrowdfunding does not exist or that there is a separating equilibrium where the high-quality�rm uses equity-based crowdfunding.20Among other things note, for example, that the proof of Lemma 5, which is crucial for

Proposition 2, relies on numerical simulations, Section 4.2.2 is not �nished and, as mentionedabove, the case when the decision about the choice of crodwfunding mode is part of the modelis not analyzed.

18

generally the most sensitive to the asymmetric information problem. Equity-based crowdfuning cannot be used as a signalling tool by a high-quality �rmsince it will always be mimicked by a low quality �rm as the share price of ahigh-quality �rm is always higher than that of a low-quality �rm. In contrast,a high-qulity can use reward-based crowdfunding. This is because a low-quality�rm may �nd it unproftable to mimick this strategy as it will be taking morerisk to achieve its threshold. This prediction has not been directly tested butis consistent with the spirit of the results found in Ahlers, Cumming, Guen-ther, and Schweizer (2015) and Mollick (2014) (that the �rm�s �nancing choicecan serve as a signal of a project�s quality). Furthermore, the entrepreneur�slarger fraction of equity is associated with a higher project quality. In our case,reward-based crowdfunding implies a higher fraction of ownership held by theentrepreneur. Ahlers et al (2015) examine the e¤ectiveness of the signals usedby entrepreneurs to induce (small) investors to commit �nancial resources inan equity-based crowdfunding context. They found that retaining equity is ane¤ective signal and can therefore strongly impact the probability of a funding�ssuccess. It is consistent with the spirit of our result that reward-based crowd-funding may be preferred by entrepreneurs of higher quality.Proposition 3 implies that high-quality projects may prefer AON over KIA.

This is consistent with the spirit of Cumming, Leboeuf and Schwienbacher(2014). They show that KIA campaigns are less successful in meeting theirfundraising goals. Also, note that the rate of success of campaigns on Kick-starter, which only uses AON, is higher than on Indiegogo.21

Proposition 1 and 4 imply that pricing and production strategies are af-fected by moral hazard issues and the costs of �nancial distress. In particular,prices can be higher and quantity produced can be lower under equity-basedcrowdfunding. This is consistent with Paakkarinen (2016) that noted that incontrast to pre-ordering, pro�t sharing may have fewer customers, but highermargins. More broadly, the point that moral hazard issues related to the entre-preneurial cost of e¤ort and the reduced equity stake are more important, underequity-based crowdfunding is consistent with Gabison (2015) and Paakkarinen(2016), which noted that equity-based crowdfunding is much more constrictedin comparison to other forms of crowdfunding.As follows from Moores (2015), the bankruptcy procedure is not clearly

de�ned in the case of a failed crowdfunding campaign, in fact, the �rm maynot even be declared bankrupt even though consumers are under the customerprotection law (at least in the case of reward-based crowdfunding). As noted inMoores (2015), further development and clari�cations in this area are helpful.Our analysis suggests that from a policy perspective higher bankruptcy costs arebene�tial for the exsitence of separating equilibria where high-quality �rms canuse reward-based crowdfunding to singal their quality and avoid being mimickedby low-quality �rms.As follows from Proposition 6, �rms should avoid crowdfunding if moral

21See, for example:http://crowdfunding.cmf-fmc.ca/facts_and_stats/how-likely-is-your-crowdfunding-

campaign-to-succeed

19

hazard considerations related to the weak ability of funders to monitor the �rm(compared to traditional �nancing from bank loans or venture capital �nancing)are very important. These results are consistent with Xu (2017) that �nds thatentrepreneurs swich between crowdfunding an bank borrowing depending on therelative costs of �nancing. Also, and perhaps more interestingly is that if weonly consider reward-based crowdfunding vs. bank �nancing (the area above line(18) in Figure 1), projects with high I and high a, i.e potentially high risk, highinvestment (novelty) and potentially high demand (a) will prefer crowdfundingvs. bank �nancing. This is also consistent with Xu (2017). Finally, we �nd that�rms should use crowdfunding for either projects with a very small demandor a very high demand. Also, �rms that use equity-based crowdfunding havea higher amount of �xed costs compared to entrepreneurs with reward-basedcrowdfunding.In Belleammey et al (2014) price discrimination is not possible in the absence

of non-monetary bene�ts, and therefore both forms of crowdfunding yield ex-actly the same outcome as seeking money from a bank or a large equity investors.Some research discovered however that the role of such non-monetary bene�ts incrowdfunding is negligeable (see, for example, Cholakova and Clarysse (2015)).In our model, there are no non-monetary bene�ts from crowdfunding but thebene�ts of crowdfunding (compared to traditional �nancing) arise from naturalfeatures of crowdfunding such as market feedback. Note that overall, the focusof most existing theoretical papers on cowdfunding has been to exploit featuresof crowdfunding like the opportunity for the entrepreneur to price discriminate.However, recent literature �nds empirically that crowdfunding also has a lot ofinformational value for entrepreneurs. Hence, our article mostly focuses on thelatter aspect of crowdfunding.Finally, note that from Lemma 1, large projects, in most cases, prefer equity-

based crowdfunding. As mentioned previously, in our case this is not due to thepresence of �nancial market imperfections but to the fact that funders can counton long-term �rm pro�ts in the case of equity-based crowdfunding. As mentionedpreviously, this result in Belleammey et al (2014) is due to the assumptionsabout community bene�ts in period 1 when �rm conducts crowdfunding. Thesebene�ts di¤er among funders in the case of a reward-based campaign so thesmall size of the crowdfunding allows the �rm to capture these di¤erences verye¢ ciently, while in the equity-based case community bene�ts are more uniformso there is no advantage of having a small scale. As follows from Paakkarinen(2016), equity-based campaigns are much larger than reward-based campaignsbut �rms select equity-based campaigns mostly for possibility of collecting alarge amounts of capital and not to select a better price discrimination approach.

9 The model extensions and robustness

Di¤erent demand functions. Our focus in this article is to analyze the roleof di¤erent market imperfections in crowdfunding. That is why we adopt arelatively simple demand function. In dynamic monopoly pricing literature

20

this approach is not unusual (see, for example, Demichelis and Tarola (2006)).Most of our results (such as Propositions 1, 2 etc.) are intuitively sound andwill hold if mathematically di¤erent demand functions are used. Alternatively,a sigi�cantly di¤erent approach of modelling the demand side can be takenwhere individual customers with di¤erent demand functions are included (see,for example, Belleammey et al (2014) and Hu, Li and Shi (2014)). This approachis often used in industrial organization or price discrimination literature. Ourfocus is on market imperfections and �nancial aspects of crowdfunding and theapproach that uses total demand functions from investors/funders (the market)is very common.22 Note also that Belleammey et al (2014) make the ad-hocassumption that crowdfunding provides an automatic bene�t to funders.Di¤erent types of moral hazard. In our model (Section 3), the entrepre-

neurial moral hazard takes place because the entrepreneur�s equity stake in the�rm is reduced while his individual e¤ort is costly and this cost is not shared.This approach is very common in �nancing literature (starting with Jensen andMeckling (1976)) and typically creates an agency cost of equity �nancing as inour paper. There are many di¤erent ways to analyze moral hazard issues, forexample, to explicitely model the entrepreneur�s level of e¤ort. This apaproachis quite common in contract literature. In �nance literature this approach wasused, for example, in Innes (1991). The result of that analysis reveals the advan-tage of debt �nancing over equity �nancing which is consistent with the spiritof our modelling where equity-based crowdfunding has a disadvantage due toentrepreneurial moral hazard. In Section 6 we again use moral hazard to com-pare crowdfunding and bank loans using the idea that bank �nancing providesbetter monitoring. This idea is standard in �nance literature (see, for example,Diamond (1984)).The distribution of types. In sections 4 and 5, which deal with asymmetric

information we use two types of �rms to illustrate the main ideas. This is alsovery typical in literature. A natural question though is whether the resultsstand if one considers a case with multiple types. Our analysis shows23 thatmost conclusions remain the same: under asymmetric information, equity-basedcrowdfunding is an inferior choice compared to reward-based crowdfunding. Inthe case of multiple types, however, one may have a semi-separating or evenpooling equilibrium where only the type with the highest cost (speaking aboutSection 4) will be indi¤erent between the two types of crowdfunding and allother types select reward-based crowdfunding. In Section 5, our analysis shows

22One can further discuss the similarity between the two approaches. One can see, forexample, that in the spirit of that literature our model can be interpretted as a case with onecustomer in each period without the possibility of product substitution between periods. Onecan see though that if substitution is allowed between periods, most results would stand sincethe period 2 product price is not less than the period 1 price in most cases so it makes nosense for this customer to wait until period 2 to purchase the product. Introducing numerouscustomers with di¤erent product valuations will de�nitely complicate the model, however,most intutions in this paper will not be a¤ected.23Proofs are available upon demand. Note that the calculations become much longer and

technically more complicated, which is very typical for multiple types games with asymmetricinformation.

21

that the results may hold even in a multiple types environment though moreresearch is required. The main implication of our analysis holds. In particular,our results show that there is no semi-separating equlibrium where the averagequality of types that choose equity-based crowdfunding or the KIA method ishigher than those that choose AON, which is consistent with our basic model.Mixed �nancing and more types of �nancing. Unlike capital structure liter-

ature, where debt/equity mix is a very common strategy (as opposite to pureequity or pure debt �nancing), simultaneously conducting di¤erent kinds ofcrowdfunding is not common. Nevertheless, if mixed �nancing is allowed inperiod 1, most results will stand. For example, if mixing bank debt and crowd-funding is allowed in period 1, as in Section 2, the results stand though thecondition (1) can be softened for a �rm if it uses equity-based crowdfunding.Similarly, Proposition 1 stands qualitatively but the formulas will be quanti-taively di¤erent. In Sections 3 and 4, a signalling equilibrium may still existwhere a high-quality �rm uses a mix of reward-based crowdfunding and a bankloan or a mix of a bank loan and AON, as in Section 4, although restrictingconditions will change quantitatively. Introducing additional �nancing strate-gies such as debt-based crowdfunding is an interesting direction. Most resutlsregarding the costs and bene�ts of di¤erent �nancing strategies found in thispaper are quite general and do not depend on introduction of more options inthe model. Quantitatively though, some conditions may change. It is de�nitelyan interesting direction for future research. Note that most existing theoreticalliterature on crowdfunding does often consider reward-based and equity-basedcrowdfunding separately from debt-based crowdfunding. One of the reasonsfor this seems to be that the founders� objectives are quite di¤erent in thesescenarios (see, for example, Hildebrand, Puri, and Rocholl (2014)).

10 Conclusions

Most existing theoretical papers on crowdfunding consider static models.24 Thispaper is one of the �rst papers that analyzes a dynamic (two-period) model ofcrowdfunding. Existing theoretical literature on crowdfunding has extensivelyfocused on such features of crowdfunding as price discrimination. This paper isone of the �rst that focuses on information aspects of crowdfunding, which ismore in the spirit of �nance literature than industrial organization literature. Inparticular, this is one of the �rst papers that obtains analytical results for modelswith asymmetric information. Most existing literature focuses more on moralhazard issues. Also, this paper is one of the �rst that analyzes the choice betweendi¤erent types of crowdfunding (reward-based vs. equity-based) and the choicebetween crowdfunding and traditional �nancing. In addition to traditional forms

24Technically there are two periods in Belleammey et al (2014) but only one period ofproduction. Periods 1 and 2 in their model di¤er in that there is pre-ordering in period 1 (orstock sales) and production takes place in period 2. In our model, pre-ordering and stage 1production happen in the same period. The presence of two production periods allows us tocapture an essential di¤erence between reward-based and equity-based crowdfunding: underequity-based crowdfunding funders can count on long-term �rm pro�ts.

22

of markets imperfections (asymmetric information, moral hazard, bankruptcycosts etc.) our model includes some other features of crowdfunding such asmarket feedback. The model provides several implications, most of which havenot been yet been tested. When asymmetric information is important, high-quality projects prefer reward-based crowdfunding. The choice of the all-or-nothing mechanism as opposed to keep-it-all can serve as a signal of a �rm�squality. Finally, crowdfunding is selected over a traditional bank loan if thedemand for the product is either very small or very large.

AppendixProof of proposition 1.In period 2, the �rm chooses p2 to maximize the entrepreneur�s pro�t

(1� �)(p2 � c)(a� p2)� e(a� p2) (19)

, which makes p2 =a+c+e�(1��)

2 .In period 1, after shares are sold, the �rm chooses p1 (q1 = a � p1) to

maximize(1� �)(p1(a� p1) +M � I � cq1)� e(a� p1) (20)

subject to M � I + cq1. Two cases are possible. If a+c+e�(1��)2 � I�Mc + a we

have p1 =a+c+e�(1��)

2 . Otherwise we have a corner solution p1 = I+ca�Mc . In

both cases, under the optimal strategy chosen by the �rm M = I + cq1.

The funders anticipate it and thereforeM and � will be connected as follows:

M = �(p1(a� p1) + (p2 � c)(a� p2)) = I + c(a� p1) (21)

Then we have:

� =I + c(a� p1)

p1(a� p1) + (p2 � c)(a� p2)Substituting this into (19) and (20) we get that the entrepreneur�s expectedpro�t over the two periods equal to:

(p1 � c� e)(a� p1)� I + (p2 � c� e)(a� p2) (22)

In the beginning of period 1, the entrepreneur selects � to maximize (22). Thecase where a+c+e�(1��)

2 < I�Mc + a is not optimal. The �rm should increase

M and � becasue of the following. (22) is concave in p1 and p1 = a+c+e2 is an

optimal p1 in (22). Further p1 =a+c+e�(1��)

2 is closer to the optimum than

p1 =I+ca�M

c . So we have p1 = p2 =a+c+e�(1��)

2 .Using the above formulas for p1 and p2, (22) can be converted into

(a� c)22

� I � e2

2(1� �)2 +e2

1� � � ea+ ec (23)

23

If � = 0, (23) will be equal to (a�c�e)22 � I. This is the same value as under

a reward-based crowdfunding scenario. When � is positive, the entrepreneur�spro�t under equity crowdfunding will be smaller since the derivative of (23) in� is negative.Proof of Proposition 2. Consider a situation where l selects reward-based

crowdfunding and h selects pro�t-sharing. If I is su¢ ciently small, we have(based on Section 2 calculations)

�h =(a� ch)2

2� I (24)

�l =(a� cl)2

2� I (25)

where �j is the equilibrium pro�t of type j (all calculations are based on thesymmetric information case for each type described in the previous section).Also we have (as follows from (21))

�h =2I + ch(a� ch)ah(a� ch)

(26)

h does not have an incentive to mimick l since, as mentioned above, in thissection asymmetric information does not concern reward-based crowdfunding.So if h chose reward-based crowdfunding it would have the same payo¤ as itwould in equilibrium: (a�ch)2

2 � I. Now suppose that l mimics h and choosesequity-based crowdfunding instead. l�s pro�t �lh then equals

�lh = (1� �h)(p1q1 + (p2l � cl)(a� p2l))

In this equation p2l = a+cl2 (as follows from Section 2) and �h is determined by

(26). Note that when l mimicks h, it has to sell a larger stake of equity in the�rm compared to the symmetric information case. Indeed if l sells equity undersymmetric information we have

�l =2I + cl(a� cl)al(a� cl)

(27)

This is smaller than (26) because cl < ch. Note that the amount of funds raisedwill be di¤erent under symmetric information. Keeping unused cash is uselessso prices and quantities will be di¤erent from the symmetric information casefor type l. More speci�cally, we have

q1 =ch(a� p1h)

cl

p1 = a� q1Indeed, I + chqh = I + ch(a � p1h) is the amount of funds raised for sellingshares. From this amount, I will cover the �xed costs for l. The remaining

24

amount ch(a�p1h) will be used to cover the variable costs of production, whichare equal to cl per unit for type l. Also

p1h =a+ ch2

It implies

�lh = (1�2I + ch(a� ch)ah(a� ch)

)((a� ch(a� ch)2cl

)ch(a� ch)

2cl+(a� cl)2

4)

This is less than (25) because cl < ch. Therefore l will not mimick h.Now consider a situation where h selects reward-based crowdfunding and l

selects pro�t-sharing. As before we have

�h =(a� ch)2

2� I (28)

�l =(a� cl)2

2� I

Suppose that h mimics l and chooses equity-based crowdfunding instead. Usingsimilar reasoning one can show that h�s pro�t �hl equals

�lh = (1�2I + cl(a� cl)al(a� cl)

)((a� cl(a� cl)2ch

)cl(a� cl)2ch

+(a� ch)2

4)

This is greater than (28) because cl < ch. Therefore h will mimick l. Thismeans that such an equilibrium does not exist.Proof of Proposition 3. Consider a situation where type l selects keep-it-all

and type h selects all-or-nothing. First we have

�1h =�(1 + �)(ah � c)2

4(29)

�1l =�(al � c)2

2(30)

where �1j is the equilibrium pro�t of type j (all calculations are based on thesymmetric information case for each type described in the previous section).Suppose that l mimics h and chooses AON. We have

�lh = �((p1h � c)(ah � p1h) + �(al � c)2

4)

where p1l = al+c2 and p1h = ah+c

2 .We have

�lh =�(ah � c)2

4+�2(al � c)2

4

Comparing this with (30) we �nd that the former is greater if

� < 2� (ah � cal � c

)2 (31)

25

and therefore type l has no incentive to deviate.Suppose that h mimics l and chooses KIA. We have

�hl = �(al � c)2

4+ �

(ah � c)24

Comparing with (29) we �nd that h does not deviate if

� > (al � cah � c

)2 (32)

Note that conditions (31) and (32) do not contradict each other. It is becausethe right side of (32) is smaller than that of (31). Indeed let x = ( al�cah�c )

2. Thenthe following makes the comparison described in the previous sentence:

x < 2� 1

x

, which always holds.Consider a situation where type h selects keep-it-all and type l selects all-

or-nothing. First we have

�1h =�(ah � c)2

2(33)

�1l =�(1 + �)(al � c)2

4(34)

where �1j is the equilibrium pro�t of type j (all calculations are based on sym-metric information case for each type described in previous section). Supposethat l mimics h and chooses KIA. We have

�lh = �(ah � c)2

4+ �

(al � c)24

This is greater than (34) because ah > al and � < 1. So a situation where typeh selects keep-it-all and type l selects all-or-nothing is not an equilibrium.Finally, consider a situation where type h selects equity-based crowdfunding.

We have

�1h =�(ah � c)2

2(35)

�1l ��(al � c)2

2(36)

(if l selects KIA, (36) holds as an equality). Suppose that l mimics h and choosesequity-based crowdfunding. l�s pro�t �lh then equals

�lh = (1� �h)(�p1l(al � p1l) + �(p2l � c)(al � p2l))

where:�h =

c

ah

26

p1l = p2l =al + c

2

It implies

�lh = (1�c

ah)�al(al � c)

2

This is greater than (36) because al < ah. Therefore l will mimick h and suchan equilibrium does not exist.Proof of Proposition 4.Consider crowdfunding. Calculations are similar to Section 4. In period 2,

the �rm chooses p2 to maximize the entrepreneur�s pro�t (1��)( p2�c�b)(a�p2), which makes p2 =

a+(b+c)= 2 .

In period 1, after the shares are sold, the �rm chooses p1 to maximize (1��)( p1(a� p1) +M � (c+ b)q1) subject to

M � (c+ b)q1 (37)

. It implies:

p1 =a+ (c+ b)=

2(38)

. The �rm�s expected pro�t in period 1 is p1(a�p1) = (a+(c+b)= )(a�(c+b)= )4 .The funders�expected earnings should cover their investment cost or:

�( (a+ (c+ b)= )(a� (c+ b)= )

4+ (a� (c+ b)= )2

4) �M (39)

Under optimal solution the conditions (37) and (39) will be bounded becausethe �rm can always make � as small as necessary to satisfy them. Then we have:

� =(c+ b)q1

(a+(c+b)= )(a�ch= )4 + (a�(c+b)= )24

=ca�(c+b)= 2 a(a�(c+b)= )

2

=c+ b

a

The entrepreneur�s expected pro�t over the two periods equals:

(1� (c+ b) a

)( a(a� (c+ b)= )

2) =

( a� c� b)22

(40)

Consider bank loan �nancing. In period 2, the �rm chooses p2 to maximize( p2 � c)(a� p2) which makes p2 = a+c=

2 . Note that this is smaller than (38).

The �rm�s expected pro�t in period 2 is (a�c= )24 .

In period 1, the �rm maximizes (p1(a� p1)+ (a�c= )24 )� c(a� p1) subject

to: p1q1 = p1(a� p1) � cq1 = c(a� p1).The solution gives us p1 =

a+c= 2 .

The �rm�s pro�t over the two periods equals

� = (a� c= )2

4+ 2(a� c= )2

4=(1 + )( a� c)2

4 (41)

27

The comparison of (40) and (41) leads to the �rst part of Proposition 4. Inparticular, (40) is decreasing, which implies that the �rm prefers crowdfundingif b is su¢ ciently small. Also, when = 1, (41) is strictly greater than (40).Proof of Proposition 5.Consider a situation where l selects reward-based crowdfunding and h selects

pro�t-sharing.Consider �rm h. Calculations are similar to Section 4. In period 2, the �rm

chooses p2 to maximize the entrepreneur�s pro�t (1��)( p2�ch)(a�p2), whichmakes p2 =

a+ch= 2 .

In period 1, after the shares are sold, the �rm chooses p1 to maximize (1��)( p1(a� p1) +M � chq1) subject to

M � cq1 (42)

. It implies: p1 =a+ch=

2 . The �rm�s expected pro�t in period 1 is p1(a�p1) = (a+ch= )(a�ch= )4 . The funders�expected earnings should cover their investmentcost or:

�( (a+ ch= )(a� ch= )

4+ (a� ch= )2

4) �M (43)

Under the optimal solution the conditions (42) and (43) will be boundedbecause the �rm can always make � as small as necessary to satisfy them. Thenwe have:

� =chq1

(a+ch= )(a�ch= )4 + (a�ch= )24

=ch

a�ch= 2

a(a�ch= )2

=ch a

The entrepreneur�s expected pro�t over the two periods equals:

(1� ch a)( a(a� ch= )

2) =

( a� ch)22

(44)

Consider �rm l. In period 2, the �rm chooses p2 to maximize ( p2�cl)(a�p2)which makes p2 =

a+cl= 2 . The �rm�s expected pro�t in period 2 is (a�cl= )2

4

In period 1, the �rm maximizes (p1(a�p1)+ (a�cl= )24 )�cl(a�p1) subject

to: p1q1 = p1(a� p1) � clq1 = cl(a� p1). The solution gives us p1 = a+cl= 2 .

The �rm�s pro�t over the two periods equals

�l = (a� cl= )2

4+ 2(a� cl= )2

4= (1 + )(a� cl= )2

4(45)

Suppose that l mimics h and chooses equity-based crowdfunding instead. l�spro�t �lh then equals

�lh = (1� �h)( p1l(a� p1l)� cl(a� p1l) + p2l(a� p2l)� cl(a� p2l))

where:�h =

ch a

(46)

28

p1l = p2l =a+ cl=

2(47)

It implies

�lh = (1�ch a) a(a� cl= )

2=( a� ch)(a� cl= )

2(48)

(48) is smaller than (45) if the following holds:

2

(1 + )< a� cl a� ch

(49)

The left side of this inequality is decreasing in and the right side is increasingin . So we have two cases. If a�cl

a�ch < 2, the condition (49) does not hold for0 < � 1 and a separating equilibrium does not exist. Otherwise it holds if is su¢ ciently high.Secondly, in order to have an equilibrium, h should not have an incentive to

switch to reward-based crowdfunding. In this case, this is a trade-o¤ betweenbankruptcy cost and the cost of moral hazard. If h switches to reward-basedcrowdfunding its payo¤ equals:

�hl = (1 + )(a� ch= )2

4

This is less than (44).Consider a situation where h selects reward-based crowdfunding and l selects

pro�t-sharing.Consider �rm l. Similarly to the above analysis we have: p1 = p2 =

a+cl= 2 ,

� = cl a and the entrepreneur�s expected pro�t over the two periods equals:

( a� cl)(a� cl= )2

(50)

Consider �rm h. We have p1 = p2 =a+ch=

2 .The �rm�s pro�t over the two periods equals

�h = (1 + )(a� ch= )2

4(51)

Suppose that h mimics l and chooses equity-based crowdfunding instead. h�spro�t �hl then equals

�hl = (1� �l)( p1h(a� p1h)� cl(a� p1h) + p2h(a� p2h)� cl(a� p2h))

It equals

�hl =( a� cl)(a� ch= )

2

This is greater than (51) because cl < ch and therefore such an equilibrium doesnot exist.

29

Proof of Proposition 6.Consider reward-based crowdfunding. In period 2, the �rm chooses p2 to

maximize (p2 � c� b)(sra� p2) which makes p2 = sra+b+c2 (all calculations are

identical to section 2.1. except that the cost equals c+ b).In period 1, the �rm maximizes (p1 � b � c)(a � p1) � I subject to p1q1 =

(p1 � c)(a� p1) � I + cq1 = I + c(a� p1).Two cases are possible. If

(a� c� b)2 < 4I (52)

then the �rm will not be able to raise enough funds to launch the production.Otherwise we have p1 = a+c+b

2 .The �rm�s pro�t over the two periods equals

�r =(a� b� c)2

4+(sra� b� c)2

4� I (53)

Consider equity-based crowdfunding. In period 2, the �rm chooses p2 to maxi-mize (1� �)(p2 � c� b)(sea� p2) which makes p2 = sea+b+c

2 .In period 1, the �rm maximizes (1 � �)(p1 � b � c)(a � p1) which makes

p1 =a+b+c2 .

The �rm�s pro�t equals

�e = (1� �)((a� b� c)2

4+(sea� b� c)2

4)

Since

�((a� b� c)2

4+(sea� b� c)2

4) = I

we have:

�e =(a� b� c)2

4+(sea� b� c)2

4� I (54)

In the case of bank loan �nancing we have p1 = p2 = a+c2 .

The �rm�s pro�t is:

�b =(a� c)22

� I (55)

Since (53) is greater than (54) we have two cases. If I is su¢ ciently small((a � c � b)2 � 4I), resulting from the comparison of (53) and (54), the �rmprefers reward-based crowdfunding to equity-based crowdfunding because sr >se. As follows from the comparison of (53) and (55), the �rm selects reward-based crowdfunding if sr is su¢ ciently large or b is su¢ ciently small. This isnot surprising given that b re�ects the degree of the moral hazard cost undercrowdfunding and sr re�ects the e¢ ciency of market feedback. Otherwise, the�rm takes a bank loan.Let us now analyze the role of demand (a) on a �rm�s decision-making. The

�rm is indi¤erent between reward-based crowdfunding and a bank loan if:

(a� b� c)24

+(sra� b� c)2

4� I = (a� c)2

2� I

30

This equation can be rewritten as:

(a� b� c)22

+(sra� b� c)2

2= (a� c)2 (56)

Since this is a quadratic equation, it implies that for any given value of I,the �rm selects equity-based crowdfunding if a is either very small or very large.Otherwise it takes a bank loan.If I is su¢ ciently large, the �rm will not be able to use reward-based crowd-

funding. As follows from the comparison of (54) and (55), the �rm selectsequity-based crowdfunding if se is su¢ ciently large or b is su¢ ciently small.Otherwise, the �rm takes a bank loan.The �rm is indi¤erent between equity-based crowdfunding and a bank loan

if:(a� b� c)2

4+(sea� b� c)2

4� I = (a� c)2

2� I

This equation can be rewritten as:

(a� b� c)22

+(sea� b� c)2

2= (a� c)2

then proceed in a similar fashion to the above analysis.

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